Question

If $E$ and $F$ are events with $P\left(E\right)\le P\left(F\right)$ and $P(E\cap F)>0$, then

• A. occurrence of $E\Rightarrow$ occurrenece of $F$
• B. occurrence of $F\Rightarrow$ occurrenece of $E$
• C. non occurrence of $E\Rightarrow$ non occurrenece of $F$
• D. none of the above imlications holds

Answer 1

The correct option is D none of the above imlications holds

Considering the set $S$.
$E$ is subset of $S$ and $F$ is subset of $S,F$ being the larger set.
As $P(E\cap F)$ is not zero then this means that $E\&F$ have something in common. But still they can be independent or independent. $E$ might be completely inside of $F$ or some very little part of $E$ is in $F$. Hence all the first three options are incorrect.